In mathematics, an indexed family is a collection of values that are associated with indexes. For example, a family of real numbers, indexed by the integers is a collection of real numbers, where each integer is associated with one of the real numbers.
Formally, an indexed family is the same thing as a function. A function with domain J and codomain X is equivalent to a family of elements of X indexed by elements of J. The only difference is that indexed families are thought of as collections instead of as functions. A value is considered to be an element of a family whenever it is an element of the image of the family's underlying function.
When a function f : J → X is treated as a family, J is called the index set of the family, the functional image f(j) for j ∈ J is denoted xj, and the mapping f is denoted {xj}j∈J or simply {xj}.
Next, if the set X is the power set of a set U, then the family {xj}j∈J is called a family of sets indexed by J .
Read more about Indexed Family: Mathematical Statement, Functions, Sets and Families, Examples, Operations On Families, Subfamily, Usage in Category Theory
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